167 research outputs found
Wie Geometrie zu einem anwendungsbezogenen und alltagsrelevanten Mathematikunterricht beitragen kann
Algorithms for finding K-best perfect matchings
AbstractIn the K-best perfect matching problem (KM) one wants to find K pairwise different, perfect matchings M1,…,Mk such that w(M1) ≥ w(M2) ≥ ⋯ ≥ w(Mk) ≥ w(M), ∀M ≠M1, M2,…, Mk. The procedure discussed in this paper is based on a binary partitioning of the matching solution space. We survey different algorithms to perform this partitioning. The best complexity bound of the resulting algorithms discussed is O(Kn3), where n is the number of nodes in the graph
Ranking robustness and its application to evacuation planning
We present a new approach to handle uncertain combinatorial optimization problems that uses solution ranking procedures to determine the degree of robustness of a solution. Unlike classic concepts for robust optimization, our approach is not purely based on absolute quantitative performance, but also includes qualitative aspects that are of major importance for the decision maker. We discuss the two variants, solution ranking and objective ranking robustness, in more detail, presenting problem complexities and solution approaches. Using an uncertain shortest path problem as a computational example, the potential of our approach is demonstrated in the context of evacuation planning due to river flooding
An Integer Network Flow Problem with Bridge Capacities
In this paper a modified version of dynamic network
ows is discussed. Whereas dynamic network flows are widely analyzed already, we consider a dynamic flow problem with aggregate arc capacities called Bridge
Problem which was introduced by Melkonian [Mel07]. We extend his research to integer flows and show that this problem is strongly NP-hard. For practical relevance we also introduce and analyze the hybrid bridge problem, i.e. with underlying networks whose arc capacity can limit aggregate flow (bridge problem) or the flow entering an arc at each time (general dynamic flow). For this kind of problem we present efficient procedures for
special cases that run in polynomial time. Moreover, we present a heuristic for general hybrid graphs with restriction on the number of bridge arcs.
Computational experiments show that the heuristic works well, both on random graphs and on graphs modeling also on realistic scenarios
Solving nonconvex planar location problems by nite dominating sets
It is well-known that some of the classical location problems with polyhedral
gauges can be solved in polynomial time by nding a fi nite dominating set, i.e. a finite set of candidates guaranteed to contain at least one optimal location.
In this paper it is fi rst established that this result holds for a much larger
class of problems than currently considered in the literature. The model for
which this result can be proven includes, for instance, location problems with
attraction and repulsion, and location-allocation problems. Next, it is shown that the approximation of general gauges by polyhedral ones in the objective function of our general model can be analyzed with regard to the subsequent error in the optimal ob jective value. For the approximation
problem two di erent approaches are described, the sandwich procedure and the greedy algorithm. Both of these approaches lead - for fixed e - to polynomial approximation algorithms with accuracy for solving the
general model considered in this paper.Dirección General de Enseñanza Superio
Note on combinatorial optimization with max-linear objective functions
AbstractWe consider combinatorial optimization problems with a feasible solution set S⊆{0,1}n specified by a system of linear constraints in 0–1 variables. Additionally, several cost functions c1,…,cp are given. The max-linear objective function is defined by f(x):=max{c1x,…,cpx: x∈S}; where cq:=(cq1,…,cqn) is for q=1,…,p an integer row vector in Rn.The problem of minimizing f(x) over S is called the max-linear combinatorial optimization (MLCO) problem.We will show that MLCO is NP-hard even for the simplest case of S⊆{0,1}n and p=2, and strongly NP-hard for general p. We discuss the relation to multi-criteria optimization and develop some bounds for MLCO
Ranking robustness and its application to evacuation planning
We present a new approach to handle uncertain combinatorial optimization problems that uses solution ranking procedures to determine the degree of robustness of a solution. Unlike classic concepts for robust optimization, our approach is not purely based on absolute quantitative performance, but also includes qualitative aspects that are of major importance for the decision maker. We discuss the two variants, solution ranking and objective ranking robustness, in more detail, presenting problem complexities and solution approaches. Using an uncertain shortest path problem as a computational example, the potential of our approach is demonstrated in the context of evacuation planning due to river flooding
Multifacility Location Problems with Tree Structure and Finite Dominating Sets
Multifacility location problems arise in many real world applications. Often, the facilities can only be placed in feasible regions such as development or industrial areas. In this paper we show the existence of a finite dominating set (FDS) for the planar multifacility location problem with polyhedral gauges as distance functions, and polyhedral feasible regions, if the interacting facilities form a tree. As application we show how to solve the planar 2-hub location problem in polynomial time. This approach will yield an ε-approximation for the euclidean norm case polynomial in the input data and 1/ε
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